A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 3, 4, 4, 1, 8, 1, 11, 9, 7, 1, 26, 7, 10, 18, 33, 1, 67, 1, 56, 37, 20, 69, 158, 1, 27, 70, 252, 1, 280, 1, 207, 402, 52, 1, 834, 133, 423, 226, 465, 1, 1132, 635, 1541, 388, 129, 1, 3377, 1, 171, 2891, 3561, 1674, 3154

Offset: 1

Author

Gus Wiseman, Jan 07 2025

Keywords

• nonn

Comments

These partitions are ranked by the odd squarefree terms of A326149.

Examples

			The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
  (2)  (9)    (12)     (15)      (18)       (20)         (21)
       (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
              (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
                       (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
                                 (9,7,2)    (11,5,4)     (12,7,2)
                                 (6,5,4,3)  (12,5,3)     (14,4,3)
                                 (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
                                 (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
                                            (9,5,4,2)    (7,5,4,3,2)
                                            (10,5,3,2)
                                            (6,5,4,3,2)
		

Crossrefs

Allowing 1's gives A379733.

The non-strict version is A379734, allowing 1's A057568.

A000041 counts integer partitions, strict A000009.

A002865 counts partitions into parts > 1.

A379666 counts partitions by sum and product, without 1's A379668.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]